CN102393687A - Method for limiting distribution and scheduling for solving machine changing problem - Google Patents

Method for limiting distribution and scheduling for solving machine changing problem Download PDF

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CN102393687A
CN102393687A CN2011102737818A CN201110273781A CN102393687A CN 102393687 A CN102393687 A CN 102393687A CN 2011102737818 A CN2011102737818 A CN 2011102737818A CN 201110273781 A CN201110273781 A CN 201110273781A CN 102393687 A CN102393687 A CN 102393687A
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equipment
processing tasks
machine
time
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CN102393687B (en
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史海波
刘昶
姚丽丽
孙德厂
韩忠华
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Shenyang Institute of Automation of CAS
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Abstract

The present invention relates to a kind of restriction allocation schedule method for solving the problems, such as to change machine, specific to occurring in semiconductor rear section manufacturing process due to changing machine phenomenon caused by different encapsulation external form optional equipments are identical, rapidoprint is different or clamping apparatus is different or changing production phenomenon. The present invention includes: step 1: initialization; Step 2: unallocated equipment collection is not sky, goes to step 3; Unallocated equipment collection is sky, goes to step 4; Step 3: statistics equipment identical with processing conditions δ h amounts to quantity
Figure DDA0000091525780000011
Calculate the permission maximum device quantity allotted that processing conditions is δ h; Step 4: to whether meet formula (2) the case where consider respectively; Step 5: processing tasks collection is sky, then terminates the distribution of this procedure, into the distribution of next process; Otherwise, step 2 is returned to. The present invention be able to solve as change machine it is unreasonable and caused by the costs such as time, cost loss and the unbalanced problem of equipment utilization; Avoid the generation for changing the frequent phenomenon of machine; Improve equipment overall utilization rate.

Description

A kind of solution changes the qualification allocation schedule method of machine problem
Technical field
The present invention relates to the qualification allocation schedule method that a kind of solution changes the machine problem, specifically to occur in the semiconductor rear section manufacture process because different encapsulation external form optional equipments are identical, rapidoprint is different or the different machine phenomenons that change that cause of clamping apparatus are perhaps changed the product phenomenon.
Background technology
Along with the raising of automated manufacturing extent of information, the organization of production of implementation informationization management has become pressing for that manufacturing industry develops rapidly with optimum decision.Production scheduling is as a key modules of organization of production management, its arrangement whether effectively with rationally directly influence economic benefit of enterprises.At present, the research of relevant production scheduling aspect has reached the period of great prosperity in theoretical circles, and especially in recent years, intelligent algorithm was produced the application in the scheduling process row, and the scheduling scheme through reasonably optimizing can make a plurality of rows produce the realization of performance index.Yet, reasons such as number of devices big, production constraint rule complicacy many owing to the product category in the actual enterprise often cause theoretic these so-called advanced algorithms infeasible.
The research of carrying out dispatching method according to the actual production scene of enterprise is to implement the key that feasible row produces system call.The present invention carries out method research to the particular schedule environment in the semiconductor production process.Aspect the research of semiconductor production planning and scheduling, domestic research unit mainly contains Tongji University and Tsing-Hua University at present.Especially people such as the Wu Qidi of Tongji University, Qiao Fei, Li Li are quite dark to the research of semi-conductor manufacturing system scheduling, and periodical literature, patent of invention, the books of delivering or publish related fields are a lot.Can know that through reading its achievement they mainly are the research that the dispatching method to semiconductor leading portion manufacturing system carries out.The manufacturing of semiconductor leading portion refers to the wafer-process part, and back segment refers to the packaging and testing part, and these two stages are generally in different enterprise production.There is the scheduling problem of can reentrying in preceding road manufacturing system, and is generally acknowledged by theoretical circles to belong to the 3rd type of scheduling problem that is different from Job-shop, Flow-shop system call.The semiconductor rear section manufacture process is relatively simple at present.But it also has the production characteristics of oneself, change like the liquid form of processing tasks in the production run, but owing to different encapsulation external form shared devices causes the equipment of many operations to exist changing producing phenomenon or the like.The existence of these problems also makes semiconductor rear section manufacturing scheduling be different from general scheduling problem.At present, to the research of semiconductor rear section manufacturing system scheduling aspect, be that periodical literature or patent of invention all do not have basically specially.The present invention describes a kind of dispatching method to " changing the machine phenomenon " in the semiconductor rear section manufacture process emphatically.Here " changing the machine phenomenon " is called " change produce phenomenon " again, refers to change on the same equipment phenomenon of processing variety, and this phenomenon regular meeting causes the replacing of rapidoprint on the equipment or clamping apparatus, thereby causes the generation of regular hour and cost price.
In document " Production Scheduling Decision model investigation ", Lu Xiangnan proposes to adopt the preferential processing rule of close person with last task character to change the product cost with reduction.This method is not done detailed introduction in the document, this method also only is to change with reduction to produce this simple target of cost as the performance evaluation index simultaneously, and reckons without the balanced problem of the utilization of resources.If only be to consider to reduce to change the product cost, then cause serious production imbalance problem to produce probably.
In the practical semiconductor production run, because the equipment price of processing semiconductor is expensive, enterprise does not generally allow device free.In order to improve plant factor, the spot dispatch personnel generally do not consider to change machine cost problem, and it is very high to change unit frequency, thereby cause a large amount of generations of production time and production cost cost.
Summary of the invention
Deficiency to prior art; The present invention provides a kind of and can solve owing to the loss and the unbalanced problem of equipment utilization that change costs such as unreasonable time of causing of machine, cost; Avoid changing the generation of the frequent phenomenon of machine, improved the equipment integral utilization factor simultaneously again and limited the allocation schedule method.
The present invention for realizing the technical scheme that above-mentioned purpose adopted is: a kind of solution changes the qualification allocation schedule method of machine problem; Be used for the operation of a plurality of processing tasks in a plurality of process equipments of dispatching semiconductors back segment manufacture process, it is characterized in that, may further comprise the steps:
Step 1: task-set to be processed is initialized as the set by way of all processing tasks of this procedure, unallocated equipment collection is initialized as the set of all process equipments of this procedure; The equipment that said unallocated equipment collection is unallocated any processing tasks;
Step 2: from task-set to be processed, the processing tasks π that selects preceding working procedure to complete the earliest i, its processing conditions is δ h, unallocated equipment collection is not empty, forwards step 3 to; Unallocated equipment collection is empty, forwards step 4 to;
Step 3: statistics and processing conditions δ hIdentical equipment amounts to quantity
Figure BDA0000091525760000021
The calculating processing condition is δ hPermission maximum device quantity allotted:
Figure BDA0000091525760000031
In the formula, M jBe the equipment sum of j procedure,
Figure BDA0000091525760000032
Processing conditions δ is satisfied in expression hProcessing tasks i in the process time of j procedure, Rounding up of expression f (x); If
Figure BDA0000091525760000034
Then never distributing equipment concentrates devices allocation of selection to give processing tasks π iProcess, the processing conditions of this equipment is set to δ h, simultaneously with the concentrated deletion of distributing equipment never of this equipment; If Then from distributing processing conditions δ hEquipment in select the devices allocation of completion the earliest to give processing tasks π i, deletion processing tasks π from task-set to be processed i, go to step 5;
Step 4: satisfying processing conditions δ hCentralized equipment select to finish the earliest the equipment of processing tasks;
Satisfying another processing conditions δ OtherCentralized equipment select to finish the earliest the equipment of processing tasks;
If satisfy
ME M k , δ other + f * R δ , j > ME M k , δ h - - - ( 2 )
Then with processing tasks π iDistribute to and satisfy processing conditions δ hCentralized equipment finish the equipment of processing tasks the earliest;
Otherwise, with processing tasks π iDistribute to and satisfy processing conditions δ OtherThe equipment that finishes processing tasks the earliest of centralized equipment, change machine production, deletion processing tasks π from task-set to be processed i, go to step 5;
In the formula (2),
Figure BDA0000091525760000037
Expression has distributed processing conditions δ hThe earliest finish time of all process equipments;
Figure BDA0000091525760000038
Expression has distributed processing conditions δ OtherThe earliest finish time of all process equipments; R δ, jBe the machine time that changes of operation j; F representes work factor, is the factor of cost price being converted into time cost, and f>=1 changes under the machine time situation equally, and when changing the machine cost price when higher, then f is bigger;
Step 5: check whether task-set to be processed is empty, if be empty, then finishes the distribution of this procedure, gets into the distribution of next process; Otherwise, turn back to step 2.
If in the said step 3
Figure BDA0000091525760000039
Processing tasks π then iInitial process time in containing the operation j that changes the machine problem and end do process time
S π i , j = E π i , j - 1 ( 3 )
E π i , j = S π i , j + t π i , j
In the formula, Expression processing tasks π iIn the beginning process time of operation j;
Figure BDA0000091525760000045
Expression processing tasks π iIn the end process time of operation j;
Figure BDA0000091525760000046
Expression processing tasks π iIn the process time of operation j;
If
Figure BDA0000091525760000047
Processing tasks π then iInitial process time in containing the operation j that changes the machine problem and end do process time
S π i , j = max { min { E M k , δ h } , E π i , j - 1 } , k = 1,2 , . . . N δ h - - - ( 4 )
E π i , j = S π i , j + t π i , j
In the formula,
Figure BDA00000915257600000411
It is δ that expression has distributed processing conditions hEquipment M kAccomplish the concluding time that all have distributed processing tasks.
In the said step 4, if satisfy formula (2), then processing tasks π iBe formula (4) shown in finishing process time the initial process time in containing the operation j that changes the machine problem;
If do not satisfy formula (2), then processing tasks π iInitial process time in containing the operation j that changes the machine problem and end are process time:
S π i , j = min { E M k , δ other } + R δ , j , if min { E M k , δ other } + R δ , j > E π i , j - 1 S π i , j = E π i , j - 1 , else - - - ( 5 )
E π i , j = S π i , j + t π i , j other≠h; k = 1,2 , . . . N δ other
In the formula,
Figure BDA00000915257600000415
Expression has distributed processing conditions not to be δ hEquipment M kAccomplish the concluding time that all have distributed processing tasks; Other representes that processing conditions is not δ hAny processing conditions; R δ, jBe the machine time that changes of the equipment among the operation j, i.e. time of conversion materials or clamping apparatus more.
Saidly containing the equipment M of the operation j that changes the machine problem kOn each processing tasks initial process time with finish process time and be:
S π M k , 1 , j = E π M k , 1 , j - 1
S π M k , 1 , j = E π M k , 1 , j - 1 + Σ i t π M k , i , j + Σ R δ h , j - - - ( 6 )
S π M k , i , j = E π M k , i - 1 , j + R δ h , j
In the formula,
Figure BDA0000091525760000051
Indication equipment M kThe processing tasks of last i processing;
Figure BDA0000091525760000052
Represent j procedure equipment M kThe initial process time of last i processing tasks, The beginning process time of then representing first processing tasks;
Figure BDA0000091525760000054
Represent j procedure equipment M kThe concluding time of last i processing tasks;
Figure BDA0000091525760000055
Represent process time;
Figure BDA0000091525760000056
Expression changes the machine time, when processing conditions is identical, do not produce when changing machine,
Comprise that also changing machine predicts in advance, judge whether to change machine, when equipment M is learnt in judgement through step 4 kIn the time of need changing machine, then as equipment M kJust can begin to change machine after accomplishing the processing tasks that has distributed, need not just to begin to change machine after next processing tasks arrival by the time, the start time that calculating changes machine is:
S R = min { E M k , δ other } if min { E M k , δ other } + R δ , j > E π i , j - 1 min { E M k , δ other } ≤ S R ≤ E π i , j - 1 - R δ , j else - - - ( 7 )
S in the formula RFor changing the machine start time the latest.
The present invention has following beneficial effect and advantage:
1. can solve owing to the loss and the unbalanced problem of equipment utilization that change costs such as unreasonable time of causing of machine, cost;
2. avoided changing the generation of the frequent phenomenon of machine;
3. improved the equipment integral utilization factor.
Description of drawings
Fig. 1 is an overview flow chart of the present invention;
Fig. 2 does not carry out equipment to limit the scheduling instance graph that distributes and change machine;
Fig. 3 does not carry out equipment to limit the scheduling instance graph that distributes and change machine arbitrarily;
Fig. 4 is the qualification allocation schedule instance graph that the present invention proposes.
Embodiment
Below in conjunction with accompanying drawing and embodiment the present invention is done further detailed description.
The overall flow figure of qualification allocation schedule method of inventing to changing the machine problem of the present invention is as shown in Figure 1.With bond sequence in the semiconductor rear section manufacture process is that example is carried out the instance explanation, and Fig. 2, Fig. 3, Fig. 4 are the figure as a result of different dispatching methods.Wherein, Fig. 2 does not limit the scheduling instance graph that distributes and change machine for not carrying out equipment; Fig. 3 does not limit the scheduling instance graph that distributes and change machine arbitrarily for not carrying out equipment; The qualification allocation schedule instance graph that Fig. 4 proposes for the present invention.Contrasting these three figure can know, use qualification allocation schedule algorithm of the present invention after, not only reduced and changed the machine number of times, reduced and changed the machine cost; The integral device utilization factor is improved simultaneously.
Table 1 is the situation such as process time, rapidoprint of end process time and bond sequence of having described the preceding working procedure of the bond sequence in the semiconductor rear section manufacture process.
Table 1
According to the data message in the table 1, the implementation process of whole dispatching method is described.Each processing tasks sorted by the preceding working procedure concluding time be: B, C, A, E, D, F, G, H, I.The equipment of bond sequence adds up to 4, is respectively bonding machine 1, bonding machine 2, bonding machine 3, bonding machine 4, and the intermediate transportation time between preceding working procedure and this procedure is not considered.
It is following that the original equipment quantity allotted limits situation:
Utilize formula (1) to calculate:
Assigning process carries out as follows:
1) according to the concluding time of preceding working procedure, at first select processing tasks B to distribute, it is the number of devices n of spun gold that statistics has been distributed processing conditions Gold=0, n Gold<N Gold, it is distributed to bonding machine 1, the rapidoprint of putting bonding machine 1 is a spun gold.Calculate according to formula (3), beginning process time of processing tasks B with finish process time and be respectively 0:15 and 0:28.5.
2) select processing tasks C to distribute, it is the number of devices n of spun gold that statistics has been distributed processing conditions Gold=1, n Gold=N Gold, then do not carry out the distribution of free device, it is distributed to the equipment-bonding machine 1 that is assigned processing tasks, utilize formula (4) to calculate, beginning process time of processing tasks C with finish process time and be respectively 0:28.5 and 0:37.5.
3) select processing tasks A to distribute, it is the number of devices n of aluminium wire that statistics has been distributed processing conditions Aluminium=0, n Aluminium<N Aluminium, then processing tasks A is distributed to bonding machine 2, the rapidoprint of putting bonding machine 2 is an aluminium wire.Calculate according to formula (3), beginning process time of processing tasks A with finish process time and be respectively 0:23 and 0:48.
4) select processing tasks E to distribute, it is the number of devices n of copper wire that statistics has been distributed processing conditions Copper=0, n Copper<N Copper, then processing tasks E is distributed to bonding machine 3, the rapidoprint of putting bonding machine 3 is a copper wire.Calculate according to formula (3), beginning process time of processing tasks E with finish process time and be respectively 0:27 and 0:42.
5) select processing tasks D to distribute, it is the number of devices n of aluminium wire that statistics has been distributed processing conditions Aluminium=1, n Aluminium<N Aluminium, then processing tasks D is distributed to bonding machine 4, the rapidoprint of putting bonding machine 4 is an aluminium wire.Calculate according to formula (3), beginning process time of processing tasks D with finish process time and be respectively 0:30 and 0:50.
6) at this moment, all free device distribute.Select processing tasks F to carry out devices allocation.Utilize formula (6) to calculate to search for; Processing conditions is that aluminium wire and the equipment of accomplishing allocating task processing the earliest are bonding machine 2, and the concluding time,
Figure BDA0000091525760000081
was 0:48; Processing conditions is not that aluminium wire and the equipment of accomplishing allocating task processing the earliest are bonding machine 1, and the concluding time, was 0:37.5.With calculating comparison: 0:37.5+1.6*7.5>0:48 in each data substitution inequality (2), satisfy inequality (2), then processing tasks F is distributed to bonding machine 2.Calculate according to formula (4), beginning process time of processing tasks F with finish process time and be respectively 0:48 and 1:03.
7) select processing tasks G to distribute.Utilize formula (6) to calculate to search for; Processing conditions is that spun gold and the equipment of accomplishing allocating task processing the earliest are bonding machine 1, and the concluding time,
Figure BDA0000091525760000083
was 0:37.5; Processing conditions is not that spun gold and the equipment of accomplishing allocating task processing the earliest are bonding machine 3, and the concluding time, was 0:42.With calculating comparison: 0:42+1.6*7.5>0:37.5 in each data substitution inequality (2), satisfy inequality (2), then processing tasks G is distributed to bonding machine 1.Calculate according to formula (4), beginning process time of processing tasks G with finish process time and be respectively 0:38 and 0:47.
8) select processing tasks H to distribute.Utilize formula (6) to calculate to search for; Processing conditions is that aluminium wire and the equipment of accomplishing allocating task processing the earliest are bonding machine 4, and the concluding time,
Figure BDA0000091525760000085
was 0:50; Processing conditions is not that aluminium wire and the equipment of accomplishing allocating task processing the earliest are bonding machine 3, and the concluding time, was 0:42.With calculating comparison: 0:42+1.6*7.5>0:50 in each data substitution inequality (2), satisfy inequality (2), then processing tasks H is distributed to bonding machine 4.Calculate according to formula (4), beginning process time of processing tasks H with finish process time and be respectively 0:50 and 1:06.
9) select processing tasks I to distribute.Utilize formula (6) to calculate to search for; Processing conditions is that aluminium wire and the equipment of accomplishing allocating task processing the earliest are bonding machine 2, and the concluding time,
Figure BDA0000091525760000087
was 1:03; Processing conditions is not that aluminium wire and the equipment of accomplishing allocating task processing the earliest are bonding machine 3, and the concluding time, was 0:42.With calculating comparison: 0:42+1.6*7.5<1:03 in each data substitution inequality (2), do not satisfy inequality (2), then processing tasks is distributed to bonding machine 3, change machine production.Utilize formula (6) to calculate and learn and change machine time S RBe 0:42; Utilize formula (5) to calculate, beginning process time of processing tasks I with finish process time and be respectively 0:49.5 and 1:04.5.Whole allocation result after the distribution is shown in accompanying drawing 4.

Claims (5)

1. a solution changes the qualification allocation schedule method of machine problem, is used for the operation of a plurality of processing tasks in a plurality of process equipments of dispatching semiconductors back segment manufacture process, it is characterized in that, may further comprise the steps:
Step 1: task-set to be processed is initialized as the set by way of all processing tasks of this procedure, unallocated equipment collection is initialized as the set of all process equipments of this procedure; The equipment that said unallocated equipment collection is unallocated any processing tasks;
Step 2: from task-set to be processed, the processing tasks π that selects preceding working procedure to complete the earliest i, its processing conditions is δ h, unallocated equipment collection is not empty, forwards step 3 to; Unallocated equipment collection is empty, forwards step 4 to;
Step 3: statistics and processing conditions δ hIdentical equipment amounts to quantity
Figure FDA0000091525750000011
The calculating processing condition is δ hPermission maximum device quantity allotted:
Figure FDA0000091525750000012
In the formula, M jBe the equipment sum of j procedure,
Figure FDA0000091525750000013
Processing conditions δ is satisfied in expression hProcessing tasks i in the process time of j procedure,
Figure FDA0000091525750000014
Rounding up of expression f (x); If
Figure FDA0000091525750000015
Then never distributing equipment concentrates devices allocation of selection to give processing tasks π iProcess, the processing conditions of this equipment is set to δ h, simultaneously with the concentrated deletion of distributing equipment never of this equipment; If
Figure FDA0000091525750000016
Then from distributing processing conditions δ hEquipment in select the devices allocation of completion the earliest to give processing tasks π i, deletion processing tasks π from task-set to be processed i, go to step 5;
Step 4: satisfying processing conditions δ hCentralized equipment select to finish the earliest the equipment of processing tasks;
Satisfying another processing conditions δ OtherCentralized equipment select to finish the earliest the equipment of processing tasks;
If satisfy
ME M k , δ other + f * R δ , j > ME M k , δ h - - - ( 2 )
Then with processing tasks π iDistribute to and satisfy processing conditions δ hCentralized equipment finish the equipment of processing tasks the earliest;
Otherwise, with processing tasks π iDistribute to and satisfy processing conditions δ OtherThe equipment that finishes processing tasks the earliest of centralized equipment, change machine production, deletion processing tasks π from task-set to be processed i, go to step 5; In the formula (2), Expression has distributed processing conditions δ hThe earliest finish time of all process equipments;
Figure FDA0000091525750000022
Expression has distributed processing conditions δ OtherThe earliest finish time of all process equipments; R δ, jBe the machine time that changes of operation j; F representes work factor, is the factor of cost price being converted into time cost, and f>=1 changes under the machine time situation equally, and when changing the machine cost price when higher, then f is bigger;
Step 5: check whether task-set to be processed is empty, if be empty, then finishes the distribution of this procedure, gets into the distribution of next process; Otherwise, turn back to step 2.
2. a kind of solution according to claim 1 changes the qualification allocation schedule method of machine problem, it is characterized in that, if in the said step 3
Figure FDA0000091525750000023
Processing tasks π then iInitial process time in containing the operation j that changes the machine problem and end do process time
S π i , j = E π i , j - 1 ( 3 )
E π i , j = S π i , j + t π i , j
In the formula,
Figure FDA0000091525750000027
Expression processing tasks π iIn the beginning process time of operation j;
Figure FDA0000091525750000028
Expression processing tasks π iIn the end process time of operation j;
Figure FDA0000091525750000029
Expression processing tasks π iIn the process time of operation j;
If
Figure FDA00000915257500000210
Processing tasks π then iInitial process time in containing the operation j that changes the machine problem and end do process time
S π i , j = max { min { E M k , δ h } , E π i , j - 1 } , k = 1,2 , . . . N δ h - - - ( 4 )
E π i , j = S π i , j + t π i , j
In the formula,
Figure FDA00000915257500000214
It is δ that expression has distributed processing conditions hEquipment M kAccomplish the concluding time that all have distributed processing tasks.
3. a kind of solution according to claim 1 changes the qualification allocation schedule method of machine problem, it is characterized in that, in the said step 4, if satisfy formula (2), then processing tasks π iBe formula (4) shown in finishing process time the initial process time in containing the operation j that changes the machine problem;
If do not satisfy formula (2), then processing tasks π iInitial process time in containing the operation j that changes the machine problem and end are process time:
S π i , j = min { E M k , δ other } + R δ , j , if min { E M k , δ other } + R δ , j > E π i , j - 1 S π i , j = E π i , j - 1 , else - - - ( 5 )
E π i , j = S π i , j + t π i , j other≠h; k = 1,2 , . . . N δ other
In the formula,
Figure FDA0000091525750000034
Expression has distributed processing conditions not to be δ hEquipment M kAccomplish the concluding time that all have distributed processing tasks; Other representes that processing conditions is not δ hAny processing conditions; R δ, jBe the machine time that changes of the equipment among the operation j, i.e. time of conversion materials or clamping apparatus more.
4. change the qualification allocation schedule method of machine problem according to claim 2 or 3 described a kind of solutions, it is characterized in that, saidly containing the equipment M of the operation j that changes the machine problem kOn each processing tasks initial process time with finish process time and be:
S π M k , 1 , j = E π M k , 1 , j - 1
S π M k , 1 , j = E π M k , 1 , j - 1 + Σ i t π M k , i , j + Σ R δ h , j - - - ( 6 )
S π M k , i , j = E π M k , i - 1 , j + R δ h , j
In the formula,
Figure FDA0000091525750000038
Indication equipment M kThe processing tasks of last i processing;
Figure FDA0000091525750000039
Represent j procedure equipment M kThe initial process time of last i processing tasks, The beginning process time of then representing first processing tasks; Represent j procedure equipment M kThe concluding time of last i processing tasks;
Figure FDA00000915257500000312
Represent process time;
Figure FDA00000915257500000313
Expression changes the machine time, when processing conditions is identical, do not produce when changing machine,
Figure FDA00000915257500000314
5. a kind of solution according to claim 1 changes the qualification allocation schedule method of machine problem, it is characterized in that, comprises that also changing machine predicts in advance, judges whether to change machine through step 4, when equipment M is learnt in judgement kIn the time of need changing machine, then as equipment M kJust can begin to change machine after accomplishing the processing tasks that has distributed, need not just to begin to change machine after next processing tasks arrival by the time, the start time that calculating changes machine is:
S R = min { E M k , δ other } if min { E M k , δ other } + R δ , j > E π i , j - 1 min { E M k , δ other } ≤ S R ≤ E π i , j - 1 - R δ , j else - - - ( 7 )
S in the formula RFor changing the machine start time the latest.
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